Suppose x were not equal to y. Then d(x,y) is not 0. Apply the definition of convergence to both x and y with to get a contradiction:
Since converges to a, there exist such that if n> , then . Since converges to b, there exist such that if n> then . Use the triangle inequality to show that those are impossible.

There is an intuitive way to consider this problem.
If then almost all of the terms of are in any ball that contains .
So if then there are two disjoint balls, one containing a the other containing b.
BUT THAT IS IMPOSSIBLE. Almost all of the terms of are in any ball that contains .